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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2025, Vol. 42 ›› Issue (1): 32-44.

• • 上一篇    下一篇

基于$\mathcal{H}$-表示求解四元数{\bf Stein}矩阵方程最小二乘问题

岳树芳1,  李  莹2,  赵建立2   

  1. 1. 莒县第三中学，山东 日照 276500
    2. 聊城大学数学科学学院矩阵半张量积理论与应用研究中心，山东 聊城 252000
  • 收稿日期:2022-01-10 接受日期:2022-12-29 出版日期:2025-02-15 发布日期:2025-04-15
  • 基金资助:
    国家自然科学基金(62176112)；山东省自然科学基金(ZR2020MA053)；聊城大学科研基金(318011921).

Solving Least Square Problem of Quaternion Stein Matrix Equation Based on $\mathcal{H}$-representation

YUE Shufang1,  LI Ying2,  ZHAO Jianli2   

  1. 1. The No.3 Middle School of Juxian, Rizhao, Shandong 276500
    2.College of Mathematical Sciences Research Center of Semi-tensor Product of Matrices: Theory and Application, Liaocheng University, Liaocheng, Shandong 252000
  • Received:2022-01-10 Accepted:2022-12-29 Online:2025-02-15 Published:2025-04-15
  • Supported by:
    The National Natural Science Foundation of China (62176112); the Natural Science Foundation of Shandong Province (ZR2020MA053); the Scientific Foundation of Liaocheng University (318011921).

摘要：

主要探讨了四元数Stein矩阵方程的最小二乘问题。首先，利用四元数矩阵的实表示方法，将四元数矩阵方程求解转变为相应实矩阵方程求解问题。其次，根据中心(斜)对称矩阵的对称结构性质，利用$\mathcal{H}$-表示提取独立元素，简化运算，给出求解四元数Stein矩阵方程最小二乘中心(斜)对称解的新方法。最后，得出该方程的最小二乘中心(斜)对称解的解集和有解的充要条件。通过数值算法给出相应算例，验证该方法和结果的有效性。

关键词: 四元数矩阵方程, 实表示矩阵, $\mathcal{H}$-表示, 中心对称矩阵, 中心斜对称矩阵

Abstract:

It mainly studies the least square solutions of quaternion Stein matrix equation. Firstly, by using the real representation method of quaternion matrix, the problem of solving quaternion matrix equation is transformed into the problem of solving corresponding real matrix equation. Secondly, according to the symmetric structural properties of the centrosymmetric (anti-centrosymmetric) matrix, using the $\mathcal{H}$-representation to extract independent elements and simplify the calculation, we give a new method for solving the least square centrosymmetric (anti-centrosymmetric) solution of the quaternion Stein matrix equation. Finally, the solution set of the least squares centrosymmetric (anti-centrosymmetric) solution of the equation and the necessary and sufficient conditions for the solution are given. The effectiveness of the method and results is demonstrated by numerical algorithms and examples.

Key words: quaternion matrix equation, real representation matrix, $\mathcal{H}$-representation, centrosymmetric matrix, anti-centrosymmetric matrix

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